   Chapter 7.3, Problem 21ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# If f : X → Y and g : Y → Z are function and g ∘ f is one-to-one, must f be one-to-one? Prove or give a counterexample.

To determine

To check:

If f:XY and  g:YZ are two functions and gf is one-to-one, then whether f must be one-to-one or not.

Explanation

Given information:

f:XY and g:YZ are two functions.

Concept used:

A function is said to be one-to-one function if distinct elements in domain must be mapped with distinct elements in co-domain.

Calculation:

Let X,Y and Z be any sets and the functions are f:XY and g:YZ.

The objective is to prove the result, if gf is one-to-one then f must be one-to-one.

To prove the function f is one-to-one, it is enough to show that if f(x1)=f(x2) then x1=x2.

For any x1 and x2 in X with f(x1)=f(x2)

As f(x1),f(x2)Y and g is a function from Y to

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